A239127 -id:A239127 - cp's OEIS frontend

A239129 a(n) = 18*n - 1, n >= 1, the second column of triangle A239127 related to the Collatz problem.

17, 35, 53, 71, 89, 107, 125, 143, 161, 179, 197, 215, 233, 251, 269, 287, 305, 323, 341, 359, 377, 395, 413, 431, 449, 467, 485, 503, 521, 539, 557, 575, 593, 611, 629, 647, 665, 683, 701, 719, 737, 755, 773, 791, 809, 827, 845, 863, 881, 899, 917, 935, 953, 971

Offset: 1

Wolfdieter Lang, Mar 13 2014

This sequence gives all ending values a(n) (in increasing order) of Collatz sequences of length 5 following the pattern (ud)^2, with u (for `up'), mapping an odd number m to 3*m+1, and d (for `down'), mapping an even number m to m/2. The last entry of this sequence is required to be odd. The first entry is also odd and is given by M(2,n) = 8*n-1 from the array A239126.

This appears as N in Example 2.2. for x=y = 2 in the M. Trümper paper on p. 7, given as a link below.

			a(1) = 17 because the Collatz sequence for M(2,1) = 8*1 - 1 = 7 from A239126 is [7, 22, 11, 34, 17] ending in the odd number 17.
a(4) = 71 with the Collatz sequence of length 5 starting with M(2,4) = 31 given by [31, 94, 47, 142, 71], ending in a(4).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Wolfdieter Lang, On Collatz' Words, Sequences, and Trees, J. of Integer Sequences, Vol. 17 (2014), Article 14.11.7.
Manfred Trümper, The Collatz Problem in the Light of an Infinite Free Semigroup, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016969 (first column), A239126, A239127.

CoefficientList[Series[(x + 17)/(1 - x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 16 2014 *)

a(n) = 18*n - 1 for n >= 1.
O.g.f.: x*(x+17)/(1-x)^2.
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: exp(x)*(18*x - 1) + 1.
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

A016969 a(n) = 6*n + 5.

Original entry on oeis.org

5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 191, 197, 203, 209, 215, 221, 227, 233, 239, 245, 251, 257, 263, 269, 275, 281, 287, 293, 299, 305, 311, 317, 323, 329, 335

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(18).
Exponents e such that x^e + x - 1 is reducible.
First differences of A141631. - Paul Curtz, Sep 12 2008
a(n-1), n >= 1, appears as first column in the triangle A239127 related to the Collatz problem. - Wolfdieter Lang, Mar 14 2014
Odd unlucky numbers in A050505. - Fred Daniel Kline, Feb 25 2017
Intersection of A005408 and A016789. - Bruno Berselli, Apr 26 2018
Numbers that are not divisible by their digital root in base 4. - Amiram Eldar, Nov 24 2022

Muniru A Asiru, Table of n, a(n) for n = 0..3000
Mark W. Coffey, Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers, arXiv:1601.01673 [math.NT], 2016.
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 949.
D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., Vol. 36, No. 3 (1935), pp. 637-649.
Amelia Carolina Sparavigna, The Pentagonal Numbers and their Link to an Integer Sequence which contains the Primes of Form 6n-1, Politecnico di Torino (Italy, 2021).
Amelia Carolina Sparavigna, Binary operations inspired by generalized entropies applied to figurate numbers, Politecnico di Torino (Italy, 2021).
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Leo Tavares, Illustration: Twin Triangular Frames.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A111863, A007310, A008588, A016921, A016933, A016945, A016957, A049452.
Cf. A050505 (unlucky numbers).
Cf. A000217.

List([0..60],n->6*n+5); # Muniru A Asiru, Nov 24 2018

[ 6*n+5: n in [0..55] ]; // Klaus Brockhaus, Jan 04 2009

6Range[0, 59] + 5 (* or *) NestList[6 + # &, 5, 60] (* Harvey P. Dale, Mar 09 2013 *)

a(n)=6*n+5 \\ Charles R Greathouse IV, Jul 10 2016

(1 to 60).map(6 *  - 1).mkString(", ") // _Alonso del Arte, Nov 23 2018

a(n) = A003415(A003415(A125200(n+1)))/2. - Reinhard Zumkeller, Nov 24 2006
A008615(a(n)) = n+1. - Reinhard Zumkeller, Feb 27 2008
a(n) = A007310(2*n+1); complement of A016921 with respect to A007310. - Reinhard Zumkeller, Oct 02 2008
From Klaus Brockhaus, Jan 04 2009: (Start)
G.f.: (5+x)/(1-x)^2.
a(0) = 5; for n > 0, a(n) = a(n-1)+6. (End)
a(n) = A016921(n)+4 = A016933(n)+3 = A016945(n)+2 = A016957(n)+1. - Klaus Brockhaus, Jan 04 2009
a(n) = floor((12n-1)/2) with offset 1..a(1)=5. - Gary Detlefs, Mar 07 2010
a(n) = 4*(3*n+1) - a(n-1) (with a(0) = 5). - Vincenzo Librandi, Nov 20 2010
a(n) = floor(1/(1/sin(1/n) - n)). - Clark Kimberling, Feb 19 2010
a(n) = 3*Sum_{k = 0..n} binomial(6*n+5, 6*k+2)*Bernoulli(6*k+2). - Michel Marcus, Jan 11 2016
a(n) = A049452(n+1) / (n+1). - Torlach Rush, Nov 23 2018
a(n) = 2*A000217(n+2) - 1 - 2*A000217(n-1). See Twin Triangular Frames illustration. - Leo Tavares, Aug 25 2021
Sum_{n>=0} (-1)^n/a(n) = Pi/6 - sqrt(3)*arccoth(sqrt(3))/3. - Amiram Eldar, Dec 10 2021
E.g.f.: exp(x)*(5 + 6*x). - Stefano Spezia, Feb 14 2025

More terms from Klaus Brockhaus, Jan 04 2009

A239126 Rectangular array showing the starting values M(n, k), k >= 1, for the Collatz operation (ud)^n, n >= 1, ending in an odd number, read by antidiagonals.

Original entry on oeis.org

3, 7, 7, 11, 15, 15, 15, 23, 31, 31, 19, 31, 47, 63, 63, 23, 39, 63, 95, 127, 127, 27, 47, 79, 127, 191, 255, 255, 31, 55, 95, 159, 255, 383, 511, 511, 35, 63, 111, 191, 319, 511, 767, 1023, 1023, 39, 71, 127, 223, 383, 639, 1023, 1535, 2047, 2047

Offset: 1

Wolfdieter Lang, Mar 13 2014

The companion array and triangle for the odd end numbers N(n, k) is given in A239127.
The two operations on natural numbers m used in the Collatz 3x+1 conjecture are here (following the M. Trümper paper given in the link) denoted by u for 'up' and d for 'down': u m = 3*m+1, if m is odd, and d m = m/2 if m is even. The present array gives all start numbers M(n, k) for the Collatz word (ud)^n = s^n (s = ud is useful because, except for the one letter word u, at least one d follows a letter u), with n >= 1, and k >= 1. Such Collatz sequences have the maximal number of u's (grow fastest).
This rectangular array is M of Example 2.2. with x=y = n, n >= 1, of the M. Trümper reference, pp. 7-8, written as a triangle by taking NE-SW diagonals. The Collatz sequence starting with M(n, k) has length 2*n+1 for each k and it ends in the odd number N(n, k) given in A239127.
The first row sequences of the array M (columns of triangle TM) are A004767, A004771, A125169, A239128, ...

			The rectangular array M(n, k) begins:
n\k     1    2    3    4     5     6     7     8     9    10 ...
1:      3    7   11   15    19    23    27    31    35    39
2:      7   15   23   31    39    47    55    63    71    79
3:     15   31   47   63    79    95   111   127   143   159
4:     31   63   95  127   159   191   223   255   287   319
5:     63  127  191  255   319   383   447   511   575   639
6:    127  255  383  511   639   767   895  1023  1151  1279
7:    255  511  767 1023  1279  1535  1791  2047  2303  2559
8:    511 1023 1535 2047  2559  3071  3583  4095  4607  5119
9:   1023 2047 3071 4095  5119  6143  7167  8191  9215 10239
10:  2047 4095 6143 8191 10239 12287 14335 16383 18431 20479
...
The triangle TM(m, n) begins (zeros are not shown):
m\n   1    2     3     4     5     6      7      8      9    10 ...
1:    3
2:    7    7
3:   11   15    15
4:   15   23    31    31
5:   19   31    47    63    63
6:   23   39    63    95   127   127
7:   27   47    79   127   191   255    255
8:   31   55    95   159   255   383    511    511
9:   35   63   111   191   319   511    767   1023   1023
10:  39   71   127   223   383   639   1023   1535   2047  2047
...
---------------------------------------------------------------------
n=1, ud, k=1: M(1, 1) = 3 = TM(1, 1), N(1,1) = 5 with the Collatz sequence  [3, 10, 5] of length 3.
n=1, ud, k=2: M(1, 2) = 7 = TM(2, 1), N(1,2) = 11 with the Collatz sequence  [7, 22, 11] of length 3.
n=4, (ud)^4, k=2: M(4, 2) = 63 = TM(5, 4), N(4,2) = 323 with the Collatz sequence  [63, 190, 95, 286, 143, 430, 215, 646, 323] of length 9.
n=5, (ud)^5, k=1: M(5, 1) = 63 =  TM(5, 5), N(5,1) = 485 with the Collatz sequence  [63, 190, 95, 286, 143, 430, 215, 646, 323, 970, 485] of length 11.

Wolfdieter Lang, On Collatz' Words, Sequences, and Trees, J. of Integer Sequences, Vol. 17 (2014), Article 14.11.7.
Manfred Trümper, The Collatz Problem in the Light of an Infinite Free Semigroup, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.
Eric Weisstein's World of Mathematics, Collatz Problem.
Wikipedia, Collatz Conjecture.

Cf. A006577, A139399, A112695, A238475, A238476, A004767, A004771, A125169, A239127, A239128.

The array: M(n, k) = 2^(n+1)*k - 1 for n >= 1 and k >= 1.
The triangle: TM(m, n) = M(n, m-n+1) = 2^(n+1)*(m-n+1) - 1 for m >= n >= 1 and 0 for m < n.
a(n) = 4*A087808(A130328(n-1)) - 1 (conjectured). - Christian Krause, Jun 15 2021

A239128 a(n) = 32*n - 1, n >= 1. Fourth column of triangle A239126, related to the Collatz problem.

Original entry on oeis.org

31, 63, 95, 127, 159, 191, 223, 255, 287, 319, 351, 383, 415, 447, 479, 511, 543, 575, 607, 639, 671, 703, 735, 767, 799, 831, 863, 895, 927, 959, 991, 1023, 1055, 1087, 1119, 1151, 1183, 1215, 1247, 1279, 1311, 1343, 1375, 1407, 1439, 1471, 1503, 1535, 1567, 1599

Offset: 1

Wolfdieter Lang, Mar 13 2014

This sequence gives all starting values a(n) (in increasing order) of Collatz sequences of length 9 following the pattern (ud)^4, with u (for `up'), mapping an odd number m to 3*m+1, and d (for `down'), mapping an even number m to m/2. The last entry of this sequence is required to be odd and it is given by 162*n-1.

This appears in Example 2.2. for x=y = 4 in the M. Trümper paper on p. 7, given as a link below.

			a(1) = 31 because the Collatz sequence following the pattern udududud is [31, 94, 47, 142, 71, 214, 107, 322, 161], with length 9, ending in the odd number N(4,1) = 161 = 162*1 - 1 from the array A239127, and 31 is the smallest positive number whose Collatz sequence follows this pattern and ends in an odd number.
a(4) = 127 with the Collatz sequence [127, 382, 191, 574, 287, 862, 431, 1294, 647] ending in N(4,4) = 647 = 32*4 - 1. 127 is the fourth smallest positive number following this pattern with odd end number.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Wolfdieter Lang, On Collatz' Words, Sequences, and Trees, J. of Integer Sequences, Vol. 17 (2014), Article 14.11.7.
Manfred Trümper, The Collatz Problem in the Light of an Infinite Free Semigroup, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A125169 (third column), A239126, A239127.

CoefficientList[Series[(31 + x)/(1 - x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 16 2014 *)
32*Range[50]-1 (* Harvey P. Dale, Jan 25 2021 *)

O.g.f.: x*(31+x)/(1-x)^2.
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: exp(x)*(32*x - 1) + 1.
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

A240222 Rectangular array giving all start values M(n, k), k >= 1, for Collatz sequences following the pattern (udd)^(n-1) ud, n >= 1, read by antidiagonals.

Original entry on oeis.org

1, 3, 1, 5, 9, 1, 7, 17, 33, 1, 9, 25, 65, 129, 1, 11, 33, 97, 257, 513, 1, 13, 41, 129, 385, 1025, 2049, 1, 15, 49, 161, 513, 1537, 4097, 8193, 1, 17, 57, 193, 641, 2049, 6145, 16385, 32769, 1, 19, 65, 225, 769, 2561, 8193, 24577, 65537, 131073, 1, 21, 73, 257, 897, 3073, 10241, 32769, 98305

Offset: 1

Wolfdieter Lang, Apr 02 2014

The companion array and triangle for the end numbers N(n, k) is given in A240223.

The two operations on natural numbers m used in the Collatz 3x+1 conjecture are here (following the M. Trümper paper given in the link) denoted by u for 'up' and d for 'down': u m = 3*m+1, if m is odd, and d m = m/2 if m is even. The present array gives all start numbers M(n, k) for Collatz sequences realizing the Collatz word (udd)^n ud = (sd)^n s (s = ud is useful because, except for the one letter word u, at least one d follows a letter u), with n >= 1, and k >= 1. The length of these Collatz sequences 3*n. For these Collatz sequences M(n, 0) = M(1, 0) = 1 and N(n, 0) = N(1, 0) = 2.

			The rectangular array M(n, k) begins:
n\k 0       1       2       3       4       5 ...
1:  1       3       5       7       9      11
2:  1       9      17      25      33      41
3:  1      33      65      97     129     161
4:  1     129     257     385     513     641
5:  1     513    1025    1537    2049    2561
6:  1    2049    4097    6145    8193   10241
7:  1    8193   16385   24577   32769   40961
8:  1   32769   65537   98305  131073  163841
9:  1  131073  262145  393217  524289  655361
10: 1  524289 1048577 1572865 2097153 2621441
...
For more columns see the link.
The triangle TM(m, n) begins (zeros are not shown):
k\n  1  2   3   4    5     6      7 ...
0:   1
1:   3  1
2:   5  9   1
3:   7 17  33   1
4:   9 25  65 129    1
5:  11 33  97 257  513     1
6:  13 41 129 385 1025  2049      1
...
For more rows see the link.
n=1, ud, k=0: M(1, 0) = 1 = TM(0, 1), N(1, 0) = 2 with the Collatz sequence [1, 4, 2] of
length 3.
n=1, ud, k=2: M(1, 2) = 5 = TM(2, 1), N(1,  2) = 8 with the Collatz sequence [5, 16, 8] of length 3.
n=2, uddud, k=0: M(2, 0) = 1 = TM(1, 2), Ne(2, 0) = 2 with the Collatz sequence [1, 4, 2, 1, 4, 2, 1, 4, 2] of length 9.

Wolfdieter Lang, Rectangular array and triangle.
Wolfdieter Lang, On Collatz' Words, Sequences and Trees, J. of Integer Sequences, Vol. 17 (2014), Article 14.11.7.
Manfred Trümper, The Collatz Problem in the Light of an Infinite Free Semigroup, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.
Eric Weisstein's World of Mathematics, Collatz Problem.
Wikipedia, Collatz Conjecture

Cf. A238475, A238476, A239126, A239127.

The array: M(n, k) = 1 + 2^(2*n-1)*k for n >= 1 and k >= 0.

The triangle: TM(m, n) = M(n,m-n+1) = 1 + 2^(2*n-1)*(m-n+1) for m+1 >= n >= 1 and 0 for m+1 < n.

A240223 Rectangular companion array to M(n,k), given in A240222, showing the end numbers N(n, k), k >= 1, for the Collatz operation (udd)^(n-1) ud, n >= 1, read by antidiagonals.

Original entry on oeis.org

2, 5, 2, 8, 11, 2, 11, 20, 29, 2, 14, 29, 56, 83, 2, 17, 38, 83, 164, 245, 2, 20, 47, 110, 245, 488, 731, 2, 23, 56, 137, 326, 731, 1460, 2189, 2, 26, 65, 164, 407, 974, 2189, 4376, 6563, 2, 29, 74, 191, 488, 1217, 2918, 6563, 13124, 19685, 2, 32, 83, 218, 569, 1460, 3647, 8750, 19685, 39368, 59051, 2

Offset: 0

Wolfdieter Lang, Apr 04 2014

The companion array and triangle for the start numbers M(n, k) is given in A240222.

For the Collatz operations u (for 'up') and d (for 'down') see the comment on A240222, also for links, especially for the M. Trümper paper.

			The rectangular array N(n, k) begins
  n\k 0      1       2       3       4       5 ...
  1:  2      5       8      11      14      17
  2:  2     11      20      29      38      47
  3:  2     29      56      83     110     137
  4:  2     83     164     245     326     407
  5:  2    245     488     731     974    1217
  6:  2    731    1460    2189    2918    3647
  7:  2   2189    4376    6563    8750   10937
  8:  2   6563   13124   19685   26246   32807
  9:  2  19685   39368   59051   78734   98417
  10: 2  59051  118100  177149  236198  295247
  ...
For more columns see the link.
The triangle TN(m, n) begins (zeros are not shown):
  m\n  1  2   3   4    5    6    7 ...
  0:   2
  1:   5  2
  2:   8 11   2
  3:  11 20  29   2
  4:  14 29  56  83    2
  5:  17 38  83 164  245    2
  6:  20 47 110 245  488  731    2
  ...
For more rows see the link.
n=1, ud, k=0: M(1, 0) = 1, N(1, 0) = TN(0, 1) = 2 with the Collatz sequence [1, 4, 2] of length 3.
n=1, ud, k=2: M(1, 2) = 5, N(1, 2) = TN(2, 1) = 8 with the Collatz sequence [5, 16, 8] of length 3.
n=2, uddud, k=0: M(2, 0) = 1, Ne(2, 0) = TN(1, 2) = 2 with the Collatz sequence [1, 4, 2, 1, 4, 2, 1, 4, 2] of length 9.

Wolfdieter Lang, Rectangular array and triangle.
Wolfdieter Lang, On Collatz Words, Sequences, and Trees, J. of Integer Sequences, Vol. 17 (2014), Article 14.11.7.
Manfred Trümper, The Collatz Problem in the Light of an Infinite Free Semigroup, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.

Cf. A238475, A238476, A239126, A239127, A240222, A016789 (first row of N), A017185 (second row of N).

The array: N(n, k) = 2 + 3^n*k for n >= 1 and k >= 0.

The triangle: TN(m, n) = N(n,m-n+1) = 2 + 3^n*(m-n+1) for m+1 >= n >= 1 and 0 for m+1 < n.

cp's OEIS Frontend

A239129 a(n) = 18*n - 1, n >= 1, the second column of triangle A239127 related to the Collatz problem.

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A016969 a(n) = 6*n + 5.

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A239126 Rectangular array showing the starting values M(n, k), k >= 1, for the Collatz operation (ud)^n, n >= 1, ending in an odd number, read by antidiagonals.

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A239128 a(n) = 32*n - 1, n >= 1. Fourth column of triangle A239126, related to the Collatz problem.

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A240222 Rectangular array giving all start values M(n, k), k >= 1, for Collatz sequences following the pattern (udd)^(n-1) ud, n >= 1, read by antidiagonals.

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A240223 Rectangular companion array to M(n,k), given in A240222, showing the end numbers N(n, k), k >= 1, for the Collatz operation (udd)^(n-1) ud, n >= 1, read by antidiagonals.

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